A signed measure is countably additive and takes at most one infinite value

Let $(X,\mathcal{A})$ be a measurable space. A signed measure on $(X,\mathcal{A})$ is a function $\nu :\mathcal{A}⟶\bar{\mathbb{R}}$ such that:

1. $\nu (\mathrm{\varnothing })=0$;
2. $\nu$ takes at most one infinite sign: either $\nu (E)\mathrm{\ne }-\mathrm{\infty }$ for every $E\in \mathcal{A}$, or $\nu (E)\mathrm{\ne }+\mathrm{\infty }$ for every $E\in \mathcal{A}$;
3. for every pairwise disjoint sequence $(E_n{)}_{n\in \mathbb{N}}$ in $\mathcal{A}$, the series on the right is defined in the extended reals and $\nu \left({\bigcup }_{n\in \mathbb{N}}{E}_n\right)={\sum }_{n=0}^{\mathrm{\infty }}\nu (E_n)$ (The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined).

The second and third clauses are load-bearing, not stylistic: together they rule out the undefined form $(+\mathrm{\infty })+(-\mathrm{\infty })$ and require the disjoint series to exist before countable additivity is asserted.

The next proposition proves the extra fact that if $\nu ({\bigcup }_n{E}_n)$ is finite, then the real series ${\sum }_n\nu (E_n)$ converges absolutely in the sense of Absolutely convergent and conditionally convergent series, and the general starting index.